3.211 \(\int \frac{\sqrt{a-b x^2}}{\sqrt{a^2-b^2 x^4}} \, dx\)

Optimal. Leaf size=64 \[ \frac{\sqrt{a-b x^2} \sqrt{a+b x^2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{\sqrt{b} \sqrt{a^2-b^2 x^4}} \]

[Out]

(Sqrt[a - b*x^2]*Sqrt[a + b*x^2]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*
Sqrt[a^2 - b^2*x^4])

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Rubi [A]  time = 0.0691381, antiderivative size = 64, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ \frac{\sqrt{a-b x^2} \sqrt{a+b x^2} \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{\sqrt{b} \sqrt{a^2-b^2 x^4}} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[a - b*x^2]/Sqrt[a^2 - b^2*x^4],x]

[Out]

(Sqrt[a - b*x^2]*Sqrt[a + b*x^2]*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*
Sqrt[a^2 - b^2*x^4])

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Rubi in Sympy [A]  time = 12.7305, size = 56, normalized size = 0.88 \[ \frac{\sqrt{a^{2} - b^{2} x^{4}} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{\sqrt{b} \sqrt{a - b x^{2}} \sqrt{a + b x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-b*x**2+a)**(1/2)/(-b**2*x**4+a**2)**(1/2),x)

[Out]

sqrt(a**2 - b**2*x**4)*atanh(sqrt(b)*x/sqrt(a + b*x**2))/(sqrt(b)*sqrt(a - b*x**
2)*sqrt(a + b*x**2))

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Mathematica [A]  time = 0.0280174, size = 67, normalized size = 1.05 \[ \frac{\log \left (\sqrt{b} \sqrt{a-b x^2} \sqrt{a^2-b^2 x^4}+a b x-b^2 x^3\right )-\log \left (b x^2-a\right )}{\sqrt{b}} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[a - b*x^2]/Sqrt[a^2 - b^2*x^4],x]

[Out]

(-Log[-a + b*x^2] + Log[a*b*x - b^2*x^3 + Sqrt[b]*Sqrt[a - b*x^2]*Sqrt[a^2 - b^2
*x^4]])/Sqrt[b]

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Maple [A]  time = 0.015, size = 54, normalized size = 0.8 \[{1\sqrt{-{b}^{2}{x}^{4}+{a}^{2}}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{-b{x}^{2}+a}}}{\frac{1}{\sqrt{b{x}^{2}+a}}}{\frac{1}{\sqrt{b}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-b*x^2+a)^(1/2)/(-b^2*x^4+a^2)^(1/2),x)

[Out]

1/(-b*x^2+a)^(1/2)/(b*x^2+a)^(1/2)/b^(1/2)*(-b^2*x^4+a^2)^(1/2)*ln(x*b^(1/2)+(b*
x^2+a)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-b*x^2 + a)/sqrt(-b^2*x^4 + a^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.276606, size = 1, normalized size = 0.02 \[ \left [\frac{\log \left (-\frac{2 \, \sqrt{-b^{2} x^{4} + a^{2}} \sqrt{-b x^{2} + a} b x -{\left (2 \, b^{2} x^{4} - a b x^{2} - a^{2}\right )} \sqrt{b}}{b x^{2} - a}\right )}{2 \, \sqrt{b}}, -\frac{\arctan \left (\frac{\sqrt{-b^{2} x^{4} + a^{2}} \sqrt{-b x^{2} + a} \sqrt{-b}}{b^{2} x^{3} - a b x}\right )}{\sqrt{-b}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-b*x^2 + a)/sqrt(-b^2*x^4 + a^2),x, algorithm="fricas")

[Out]

[1/2*log(-(2*sqrt(-b^2*x^4 + a^2)*sqrt(-b*x^2 + a)*b*x - (2*b^2*x^4 - a*b*x^2 -
a^2)*sqrt(b))/(b*x^2 - a))/sqrt(b), -arctan(sqrt(-b^2*x^4 + a^2)*sqrt(-b*x^2 + a
)*sqrt(-b)/(b^2*x^3 - a*b*x))/sqrt(-b)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{a - b x^{2}}}{\sqrt{- \left (- a + b x^{2}\right ) \left (a + b x^{2}\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-b*x**2+a)**(1/2)/(-b**2*x**4+a**2)**(1/2),x)

[Out]

Integral(sqrt(a - b*x**2)/sqrt(-(-a + b*x**2)*(a + b*x**2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-b x^{2} + a}}{\sqrt{-b^{2} x^{4} + a^{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-b*x^2 + a)/sqrt(-b^2*x^4 + a^2),x, algorithm="giac")

[Out]

integrate(sqrt(-b*x^2 + a)/sqrt(-b^2*x^4 + a^2), x)